相关论文: Tokens: An Algebraic Construction Common in Combin…
We provide a new way to represent numerical semigroups by showing that the position of every Ap\'ery set of a numerical semigroup $S$ in the enumeration of the elements of $S$ is unique, and that $S$ can be re-constructed from this…
We present an algebraic study of a kind of quantum systems belonging to a family of superintegrable Hamiltonian systems in terms of shape-invariant intertwinig operators, that span pairs of Lie algebras like $(su(n),so(2n))$ or…
New features of systems with non-trivial topology such as fractional quantum numbers, inequivalent quantizations, good operators, topological anomalies, etc. are described in the framework of an algebraic quantization procedure on a group.…
We review the basic properties of paired operators and their adjoints, the transposed paired operators, with particular reference to commutation relations, and we study the properties of their kernels, bringing out their similarities and…
Recent progress in string theory has led to a reformulation of quantum-group polynomial invariants for knots and links into new polynomial invariants whose coefficients can be understood in topological terms. We describe in detail how to…
Quantum physics has revealed many interesting formal properties associated with the algebra of two operators, A and B, satisfying the partial commutation relation AB-BA=1. This study surveys the relationships between classical combinatorial…
We establish analogues in the context of group actions or group representations of some classical problems and results in additive combinatorics of groups. We also study the notion of left invariant submodular function defined on power sets…
For $l,n \in \mathbb{N}$ we define tonal partition algebra $P^l_n$ over $\mathbb{Z}[\delta]$. We construct modules $\{ \Delta_{\underline{\mu}} \}_{\underline{\mu}}$ for $P^l_n$ over $\mathbb{Z}[\delta]$, and hence over any integral domain…
A quotient construction defines an abstract type from a concrete type, using an equivalence relation to identify elements of the concrete type that are to be regarded as indistinguishable. The elements of a quotient type are…
Full formal descriptions of algorithms making use of quantum principles must take into account both quantum and classical computing components and assemble them so that they communicate and cooperate. Moreover, to model concurrent and…
We define and analyze Toeplitz operators whose symbols are the elements of the complex quantum plane, a non-commutative, infinite dimensional algebra. In particular, the symbols do not come from an algebra of functions. The process of…
The existing relation between the tomographic description of quantum states and the convolution algebra of certain discrete groupoids represented on Hilbert spaces will be discussed. The realizations of groupoid algebras based on qudit,…
A quantum set is defined to be simply a set of nonzero finite-dimensional Hilbert spaces. Together with binary relations, essentially the quantum relations of Weaver, quantum sets form a dagger compact category. Functions between quantum…
Single-cell sequencing technology maps cells to a high-dimensional space encoding their internal activity. Recently-proposed virtual cell models extend this concept, enriching cells' representations based on patterns learned from…
The Dunkl operators associated to a dihedral group are a pair of differential-difference operators that generate a commutative algebra acting on differentiable functions in $\mathbb{R}^2$. The intertwining operator intertwines between this…
Systems of equations with sets of integers as unknowns are considered. It is shown that the class of sets representable by unique solutions of equations using the operations of union and addition $S+T=\makeset{m+n}{m \in S, \: n \in T}$ and…
We describe general methods for enumerating subsemigroups of finite semigroups and techniques to improve the algorithmic efficiency of the calculations. As a particular application we use our algorithms to enumerate all transformation…
We introduce the quasi-partition algebra $QP_k(n)$ as a centralizer algebra of the symmetric group. This algebra is a subalgebra of the partition algebra and inherits many similar combinatorial properties. We construct a basis for…
In Transformer architectures, tokens\textemdash discrete units derived from raw data\textemdash are formed by segmenting inputs into fixed-length chunks. Each token is then mapped to an embedding, enabling parallel attention computations…
We give an axiomatic formulation of quantum structures like semilogics and quasilogics which generalize the boolean semirings of events and fuzzy logics. The notions of distributions, states, representations observables and semiobservables…